The Robin problems for the coupled system of reaction–diffusion equations

This article investigates the local well-posedness of Turing-type reaction–diﬀusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas uniﬁed transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, deﬁned by the uniﬁed transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem.


Introduction
The diffusion equation is a widely used concept in contemporary science, employed to describe various phenomena in physics, chemistry, and biology.In 1952, Alan Turing used this equation to explain natural patterns in a ground-breaking way.In the realm of physics, the heat equation is a prominent example of a diffusion equation.Joseph Fourier developed it in 1822 to model the diffusion of heat within a specific area.This classical parabolic partial differential equation is a significant subject in pure mathematics and has been extensively researched.The study of the heat equation is a cornerstone of the field of partial differential equations.Additionally, considering the heat equation on Riemannian manifolds leads to many geometric applications.In the field of biology, the classical Lotka-Volterra equation system is another example of a diffusion equation system.This model provides a framework for understanding variations in predator and prey populations.In conclusion, the diffusion equation has numerous scientific applications, and its substantial contribution to the advancement of human knowledge requires further research and development.
We now present recent articles that address the existence, uniqueness, and wellposedness of solutions related to the reaction-diffusion equations.Slavík, Stehlík, and Volek [20] examine issues concerning lattice reaction-diffusion equations, utilizing maxi-mum principles to establish results of existence, uniqueness, and continuous dependence.They establish both the local existence and global uniqueness of bounded solutions, as well as the continuous dependence of solutions on the underlying time structure and initial conditions.The weak maximum principle is applied to prove the global existence of solutions.Finally, the authors provide the strong maximum principle, revealing an intriguing dependence on the time structure.Xu, Lian, and Nin [22] study nonlinear parabolic systems with power-type source terms, dividing the study into three cases based on initial energy considerations.In the low initial energy scenario, they use the Galerkin method and the concave function method to establish the global existence and finite-time blowup of the solution.For the critical initial energy case, the global solution, the blowup solution, and the asymptotic behavior are proved by scaling the initial data.In the high initial energy case, the authors explore the potential for both the global existence and the finitetime blowup by finding the corresponding initial data with arbitrarily high initial energy and then provide proof of the global existence.Palencia and Redondo [18] investigate the existence, uniqueness, and positivity conditions for a cooperative system formulated with high-order diffusion.They demonstrate the oscillatory behavior of self-similar solutions and characterize regions of positivity for a class of high-order cooperative systems without advection.Palencia, Rahman, and Redondo [17] analyzed a Fisher-KPP nonlinear reaction equation within a framework involving higher-order diffusion and the presence of an advection term.Palencia and Rahman [16] proposed a new model to describe the behavior of flames driven by temperature and pressure variables.They used the p-Laplacian operator in flame propagation, making their model applicable to a wide range of diffusion-driven domains, and they proved the uniqueness and boundedness of the weak solution and the existence of a minimum traveling-wave speed.Palencia [15] studied a reaction-diffusion problem involving high-order operators, nonlinear advection, and Fisher-KPP reaction terms.The author introduced a novel extended operator to study the reaction within the open domain R n but depart from a sequence of bounded domains.Regularity, existence, and uniqueness analyses of the solutions were performed using semigroup theory.Morgan and Tang [11] investigate the global existence of classical solutions for volume-surface reaction-diffusion systems with mass control.They introduce a novel family of L p -energy functions and utilize a general assumption known as the intermediate sum condition to establish the global existence of classical solutions.Himonas, Mantzavinos, and Yan [8] use the unified transform method to prove the local well-posedness of the reaction-diffusion equations with the Dirichlet boundary conditions.
In contemporary scientific research, coupled systems with Robin boundary conditions are extensively applied.Well-posedness ensures the equation models' reliability and predictive accuracy in various fields, making it critical for scientific research, engineering applications, and decision making.Based on our current understanding derived from relevant studies on reaction-diffusion equations, we have looked at the local well-posedness of the coupled system of reaction-diffusion equations with Robin boundary conditions in this article.

Main results
The occurrence of patterns is ubiquitous in the natural world, appearing in diverse forms such as stripes and spots on animals, intricate branching patterns in leaves, and the remarkable structural diversity observed in both biological and nonbiological systems.The investigation and comprehension of the underlying mechanisms of pattern formation have been central subjects of scientific inquiry.In 1952, Alan Turing [21] made a substantial contribution to this field by studying Turing-type reaction-diffusion equations: where A and B are positive constants in R, and u and v are morphogen concentrations, F and G describe the interrelation between morgens, Au xx and Bv xx can move randomly with diffusivities A and B. His research suggested a possible connection between the mathematical models described by these equations and the actual processes of pattern formations.Turing's work laid the groundwork for studying the mathematical aspects of pattern emergences in various systems.Following Turing's pioneering efforts, numerous articles and studies have been devoted to advancing our understanding of Turing-type reactiondiffusion equations and their implications for pattern formations.These equations have been applied in a variety of fields, providing insights into the emergence of patterns in natural systems.References to these articles and books are available for further explorations [1,7,9,10,[12][13][14]19].
In this study, we investigate the local well-posedness of the initial boundary value problem (IBVP) given by equation (1).Our proof of the local well-posedness of (1) consists of three steps.In the first step, we replace the nonlinearities u 2 + cuv and v 2 + duv by the forcings and use the Unified Transform Method (UTM) to solve the corresponding linear IBVPs.introduced the UTM and its applications.)The second step involves deriving linear estimates using the UTM formula with data and forcing in appropriate spaces.The third step shows that the iteration map defined by the UTM formula, with the forcing replaced by the nonlinearity, is a contraction map in an appropriate solution space.Finally, the uniqueness of the solution for the IBVP (1) is established by the contraction mapping theorem.In addition, we prove the local Lipschitz continuity of the data-to-solution map, thereby confirming the local well-posedness of the IBVP (1).Now, we provide an overview of the Sobolev space.The Sobolev spaces H s x (0, ∞) and H (2s-1)/4 t (0, T) are derived as restrictions of their counterparts over the entire real line, following the general definition: For s ∈ R, Sobolev spaces H s (R) consist of all tempered distributions F with the finite norm , where F(ξ ) is the Fourier transform defined by Furthermore, for an open interval in R, the Sobolev space H s ( ) is defined as By solving the forced linear Robin IBVP via the Fokas method, it leads us to the following Fourier transform.Definition 1.1 (Fourier transform on the half-line) For a test function φ(x) defined on (0, ∞), its half-line Fourier transform is given by the formula where k ∈ C and (k) ≤ 0. The notations (k) and (k) represent the imaginary and real parts of k.
Remark 1.2 For (2), it is obvious that if φ is an integrable function on (0, ∞), we observe that φ(k) is well defined for (k) ≤ 0. In fact, in the context of a more appropriate space L 2 (0, ∞), it is possible to define the half-line Fourier transform.Subsequently, the function φ in L 2 (0, ∞) can be extended to the entire real line by assigning φ(x) = 0 for x < 0, yielding a function in L 2 (R).Furthermore, the half-line Fourier transform of φ can be defined using the same formula used for the Fourier transform of φ and its extension to the real line.It follows that the formula for the inverse can also be obtained, which is the inverse Fourier transform on the real line.
Let us begin by outlining the first step of our approach to solving the problem for the associated forced linear equation: According to the UTM formulation, the solution to (3) is denoted by Figure 1 The region D + and its positively oriented boundaries ∂D where is the Fourier transform of f (x, y) with respect to x, and D + represents the domain in the complex k plane shown in Fig. 1.
For ease of calculation and presentation, we use the following notations.
Remark 1.3 For two quantities A and B depending on one or several variables, we express A B if there exists a positive constant c such that A ≤ cB.If A B and B A, then we denote A B. Now, we delineate the second step, which involves estimating the Hadamard norm of the UTM solution formula S[u 0 , g 0 ; f ] in (4) by the Sobolev norms of the data and an appropriate norm of the forcing.More precisely, we derive the following linear estimate.
Finally, our goal is to prove the uniqueness of the solution for (1) and establish that the data-to-solution is locally Lipschitz continuous.Therefore, for s > 1/2 and 0 < T * ≤ T < 1, we define two Banach spaces X and D as The data space which has the norm defined by + g 0 Then, using the above definitions, we give the main result of this work.
In Sect.2, we study a reduced pure IBVP for the linear reaction-diffusion equation to derive Theorem 2.1 and Theorem 2.4, which help to prove Theorem 1.4.In Sects. 3 and 4, we provide the proofs of Theorem 1.4 and Theorem 1.5, respectively.

The reduced pure IBVP for the linear reaction-diffusion equation
In this section, we analyze a basic Robin problem associated with the linear reactiondiffusion equation to establish Theorem 2.1 and Theorem 2.4.These theorems serve as crucial tools for estimating linear IBVPs (III) and (IV) in Sect.3.

Reduced pure IBVP
We start by considering the fundamental linear reaction-diffusion equation IBVP on the half-line.This corresponds to the homogeneous IBVP with zero initial data and nonzero boundary data.
In addition, we assume that the boundary data g ∈ H (2s-1)/4 t (R) is a time-dependent test function with compact support in the interval [0, 2].This particular problem, known as the reduced pure IBVP, can be formulated as follows: Taking advantage of the compact support of g, we express its time transformation over the interval (0, 2) as a full Fourier transform, denoted by By the UTM formula, the solution of the reduced pure IBVP (7) is for all x ∈ [0, ∞) and t ∈ R. Now, we compute (8).First, we calculate and on the other hand g k e i π 4 2 , t e i π 4 dk , Therefore, we can rewrite (8) as where and In the following result, we estimate the solution (9) in the Hadamard space.
Theorem 2.1 (Estimates for the pure linear IBVP on the half-line) For 1/2 < s < 3/2 and the boundary data test function g ∈ H (2s-1)/4 t (R) is compactly supported in the interval [0, 2].Then, the solution of the reduced pure IBVP (7), which satisfies the following Hadamard space estimates: time estimate: where C s = C(s) > 0 is a constant depending on s.
Proof First, we start with the proof of the space estimate (11).We can derive the inequalities (13) and but we will only present the proof for the inequality (13).Since the estimation processes of ( 13) and ( 14) are similar, we can use a process analogous to the proof of ( 13) to obtain (14).Therefore, by ( 9), (13), and ( 14), we establish the equation for the space estimate (11): . Now, we begin the proof of the inequality (13).We use the physical space definition of the H s x (0, ∞) norm: where 0 < β < 1 and s = sβ ∈ Z + ∪ {0}.The fractional norm • β is defined by dζ dx, ∀β ∈ (0, 1).
We require the following two lemmas in [6] to assist us in proving the inequality (13) under Cases (I)-(III).
We establish the validity of the inequality (13) under these three cases.Thus, we derive the space estimate for w 1 (13): Additionally, we can use a process similar to the proof of equation ( 13) to obtain the space estimate for w 2 (14): Then, we can obtain the space estimation (11).Finally, we start with the proof of the space estimate (12).Now, we calculate (9): where a = e iπ /4 .Hence, by equation ( 22), we infer that the temporal Fourier transform of w is given by w(x, t) Therefore, we obtain the following inequality: . Hence, we obtain the time estimate (12): . Thus, we conclude the demonstration of Theorem 2.1.
Hence, for IBVP (23), we can obtain the following two inequalities (space estimate and time estimate): , by Theorem 2.1 and time estimate: where C s = C(s) > 0 is a constant depending on s.
3 Proof of forced linear IBVP estimates (Theorem 1.4) In this section, we aim to prove Theorem 1.4 by decomposing the Robin problem for the forced linear reaction-diffusion equation into four simpler problems.In particular, two of these problems are linear initial value problems (IVPs), and they are estimated directly; their proofs can be found in [8].The remaining two problems are IBVPs, and their estimates are obtained using the theorems presented in Sect. 2.

Decomposition into simple problems
To establish the proof of Theorem 1.4, we begin the process by decomposing the forced linear IBVP (3) into the superposition of the following problems.
(I) The homogeneous linear initial value problem: where with the solution to IVP (27) given by the Duhamel formula where U 0 (ζ ) is the Fourier transform with respect to the spatial variable, i.e., (II) The forced linear IVP with zero initial condition: where The solution to IVP (30) is given by which is found by the whole-line Fourier transform where F is the Fourier transform of F with respect to x.

The estimates for the linear IVPs
In this subsection, we prove Theorem 3.
where C s = C(s) > 0 is a constant depending on s.

Theorem 3.2 ([8]. Sobolev-type estimates for the homogeneous linear IVP (II))
The solution W = S[0; F] of the forced linear IVP (30) given by the formula (32) admits the following space and time estimates: where C s = C(s) > 0 is a constant depending on s.
The proofs of the above two Theorems are provided in [8].
Proof We will use Lemmas 3.4-3.7 to prove Theorem 3.3.In the first step, we will use Lemma 3.4 and Lemma 3.5 to prove (42) of Theorem 3.3.In the second step, we will use Lemma 3.6 and Lemma 3.7 to prove (43) of Theorem 3.3.For 1/2 < s < 3/2, we assume m = (2s -1)/4, then 0 < m < 1 2 .Based on equations ( 15) and (29), we obtain the following two equations: where the fraction norm • m is defined by Therefore, we must estimate U x (x) L 2 t (0,T) and U x (x) m to obtain (42).In the following lemma, we provide the estimate for U x (x) L 2 t (0,T) .
Lemma 3.4 For 1/2 < s < 3/2.The solution U = S[U 0 ; 0] of the linear IVP (27) given by the formula (29) admits the following estimate: Proof To estimate U x (x) L 2 t (0,T) , we obtain the following inequality: , (by Lemma 2.3).Now, we calculate equations (A) and (B) to obtain the following two inequalities: Hence, we obtain the inequality (44): Next, we provide the estimate for U x (x) m in the following lemma.given by the formula (29) admits the following estimate, Proof To estimate U x (x, t) m , we obtain the following inequality: (iξ )e iξ x e -ξ 2 (t+ζ )e -ξ 2 t U 0 (ξ ) dξ . Now, we calculate equations (C) and (D) to obtain the following two inequalities: Therefore, we obtain the inequality (45): Now, we can prove (42) of Theorem 3.3.By Lemma 3.4 and Lemma 3.5, we obtain the following inequality: Hence, we obtain the inequality (42): Next, we will use Lemma 3.6 and Lemma 3.7 to prove (43) of Theorem 3.3.For 1/2 < s < 3/2, we assume m = (2s -1)/4, then 0 < m < 1 2 .Based on equations ( 15) and (32), we obtain that where the fraction norm • m is defined by Therefore, we must estimate W x (x) L 2 t (0,T) and W x (x) m to obtain (43).In the following lemma, we provide the estimate for W x (x) L 2 t (0,T) .
Lemma 3.6 For 1/2 < s < 3/2.The solution W = S[0; F] of the forced linear IVP (30) given by the formula (32) admits the following estimate: Proof To estimate W x (x) L 2 t (0,T) , we obtain the following inequality: Hence, we obtain the inequality (46): Next, we provide the estimate for W x (x) m in the following lemma.
Lemma 3.7 For 1/2 < s < 3/2 and m = (2s -1)/4.The solution W = S[0; F] of the forced linear IVP (30) given by the formula (32) admits the following estimate: Proof To estimate W x (x, t) m , we obtain the following inequality: . Now, we estimate (F) to obtain the following inequality: , by the Sobolev Embedding Theorem dζ dt by Minkowski's Integral Inequality .
Therefore, we obtain the following inequality: Next, we estimate (E) to obtain the following inequality: 6 e -2r 2 dr By ( 50) and (51), we derive the following inequality: Hence, we obtain the inequality (47): Now, we can prove (43) of Theorem 3.3.By Lemma 3.6 and Lemma 3.7, we obtain the following inequality: Therefore, we obtain the inequality (43): Hence, we finish the proof of Theorem 3.3.Now, we begin the proof of Theorem 1.4.For 1/2 < s < 3/2, to prove Theorem 1.4, we need to estimate the equation (35): to obtain its space and time estimates.
Hence, we obtain the following time estimates of (35): Accoring to (54) and (55), we obtain the inequality (5): where C s = C(s) ≥ max{C s , C s } is a constant depending on s.Hence, we have concluded the proof of Theorem 1.4.
To discuss the local well-posedness of (1), it is essential to consider the following IBVPs: Since the estimation processes of ( 3) and ( 56) are similar, we can also obtain the following result: where d s > 0 is a constant depending on s.
In the upcoming section, we will establish the proof for Theorem 1.5.Inequalities ( 5) and (57) play a crucial role in the proof of Theorem 1.5.

The local well-posedness of the coupled system of the reaction-diffusion equations (the proof of Theorem 1.5)
In this section, we define the iteration map.Subsequently, in Lemma 4.1 and Lemma 4.2, we prove that the iteration map is a contraction and onto a closed ball.We then utilize the contraction mapping theorem to establish the uniqueness of the solution.In the following Lemma 4.3, we prove that the data-to-solution is locally Lipschitz continuous.Therefore, we can use Lemma 4.1, Lemma 4.2, and Lemma 4.3 to complete the proof of Theorem 1.5.Now, we define the iteration map: which is obtained from the UTM solution formulas: t 0 e k 2 y p(k, y) dy dk.
More precisely, we have i.e., the iteration map We will demonstrate that our iteration map (58) is a contraction in the complete metric space Next, we consider a closed ball B(0, r) = {(u, v) ∈ X : (u, v) X ≤ r}, where C * s = max{C s , d s } and r = 2C * s (u 0 , v 0 , g 0 , h 0 ) D .In the following lemma, we determine the constraint on T * such that E is onto B(0, r).Lemma 4.1 For C * s = max{C s , d s } and r = 2C * s (u 0 , v 0 , g 0 , h 0 ) D .The iteration map E(u, v) is onto B(0, r), when the following condition on T * holds: Proof For (u, v) ∈ B(0, r), we obtain the following inequality,  4 (2 + |c| + |d|) 2 (u 0 , v 0 , g 0 , h 0 ) 2 D is satisfied.Hence, we obtain that E is onto B(0, r) when T * satisfies (60).
Hence, when we set

Theorem 3 . 1 (
[8].Estimates for the homogeneous linear IVP (I)) The solution U = S[U 0 ; 0] of the linear IVP (27) given by the formula (29) admits the following space and time estimates: